Finite Group Databasebeta

← #13358213

Subgroups — #13358213

Order 1792.

730 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2922C211yesmaximal, characteristic, complemented
2#11739951C211yesmaximal, characteristic
3#11735900C211yesmaximal, characteristic, complemented
4#11729397C211yesmaximal, characteristic, complemented
5#11728292C211yesmaximal, characteristic, complemented
6#11728291C211yesmaximal, characteristic, complemented
7#11728290C211yesmaximal, characteristic, complemented
8#11728289C211yesmaximal, characteristic, complemented
9#11727754C211yesmaximal, characteristic
10#11727753C211yesmaximal, characteristic
11#11727752C211yesmaximal, characteristic, complemented
12#11727751C211yesmaximal, characteristic, complemented
13#11727627C211yesmaximal, characteristic, complemented
14#11727620C211yesmaximal, characteristic, complemented
15#11727503C211yesmaximal, characteristic, complemented
16C4×C7×#41C2211yescharacteristic
17C2×#6513C2211yescharacteristic
18C4×#1157C2211yescharacteristic
19C7×#433—22no—
20C7×#431—22no—
21C7×#424C2211yescharacteristic
22C7×#424C2211yescharacteristic
23C7×#408C2211yescharacteristic
24C7×#408C2211yescharacteristic
25C4×C7×M16C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27C2×#6411C2211yescharacteristic
28C2×#6411C2211yescharacteristic
29#90005C2211yescharacteristic
30C2×#6399C2211yescharacteristic
31C2×#6399C2211yescharacteristic
32C2×#6396C2211yescharacteristic
33C2×#6396C2211yescharacteristic
34#89882C2211yescharacteristic
35#89882C2211yescharacteristic
36#89874C2211yescharacteristic
37#89874C2211yescharacteristic
38C2×#6389C2211yescharacteristic
39C2×#6389C2211yescharacteristic
40C2×C8×Dic7C2211yescharacteristic
41C2×C8×Dic7C2211yescharacteristic
42#89802C412yescomplemented
43#89798C412yescomplemented
44#89790C412yescomplemented
45#89786C412yescomplemented
46#89774C2211yescharacteristic
47#89770C2211yescharacteristic
48#89704C2211yescharacteristic
49C4×#1136C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51#89462—22no—
52#89457—22no—
53#89353C2211yescharacteristic
54#89353C2211yescharacteristic
55#89350C2211yescharacteristic
56#89350C2211yescharacteristic
57#89327C2211yescharacteristic
58#89327C2211yescharacteristic
59#21296—71nomaximal, Sylow
60C2×C7×#41C2311yescharacteristic
61C2×#1157C2311yescharacteristic
62C2×C7×M16C2311yescharacteristic
63C2×C7×M16C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C4×C7×Q8C2×C412yes—
67C4×C7×D4—22no—
68C4×C7×D4C2×C412yes—
69C7×#117—21no—
70C7×#117C2311yescharacteristic
71C2×C42×C7—21no—
72C2×C42×C7C2311yescharacteristic
73#6513—22no—
74#6513—22no—
75C2×#1144C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77C2×#1142C2×C411yescharacteristic
78C2×#1141C2×C411yescharacteristic
79C4×C22⋊D7—22no—
80#6489C2311yescharacteristic
81C2×C4×Dic7C2311yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C2×#1136C2311yescharacteristic
84C2×#1136C2311yescharacteristic
85C22×C7⋊C8C2311yescharacteristic
86C22×C7⋊C8C2311yescharacteristic
87C4×D28C2×C412yes—
88#6435—21no—
89C42×D7—21no—
90C4×Dic14C2×C412yes—
91C7×C4⋊C8—22no—
92C7×C4⋊C8—22no—
93C7×C22⋊C8—22no—
94C7×C22⋊C8—22no—
95C7×#97—21no—
96C4×C7×C8—21no—
97C4×C7×C8D412yes—
98C4×C7×C8D412yes—
99#6411—22no—
100#6411—22no—
101#6409C2×C412yes—
102#6406C2×C412yes—
103C7⋊#95—22no—
104#6399—22no—
105#6399—22no—
106#6396—22no—
107#6396—22no—
108(C2×D7)⋊C8—22no—
109(C2×D7)⋊C8—22no—
110#6389—22no—
111#6389—22no—
112Dic7⋊C8—22no—
113Dic7⋊C8—22no—
114C8×Dic7—22no—
115C8×Dic7—22no—
116#6384C2×C412yes—
117#6383C2×C412yes—
118#6382C2×C412yes—
119#6381C2×C412yes—
120#6377—21no—
121#6377C2×C412yes—
122#6377C2×C412yes—
123C4×C7⋊C8—21no—
124#2943—71no—
125#2922—71no—
126#2922D711yescharacteristic, complemented
127#1975—71no—
128#1974—71no—
129#1946—71no—
130#1945—71no—
131#1938—71no—
132#1938—71no—
133#1937—71no—
134#1937—71no—
135#1778—71no—
136#1759—71no—
137#1759—71no—
138#1726—71no—
139C7×#41—21no—
140C7×#41—21no—
141C7×#41—21no—
142C7×#41—21no—
143C2×C7×Q8C22×C411yescharacteristic
144C2×C7×D4—21no—
145C2×C7×D4C22×C411yescharacteristic
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7#4112yes—
149C22×C4×C7C2411yescharacteristic
150C2×C22⋊D7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153#1157—21no—
154#1157—21no—
155#1157—21no—
156#1157—21no—
157C2×D28C22×C411yescharacteristic
158C2×C4×D7—22no—
159C2×C4×D7—21no—
160C2×Dic14C22×C411yescharacteristic
161C7×M16—21no—
162C7×M16—21no—
163C7×M16—21no—
164C7×M16—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8C2×D412yes—
172C2×C7×C8C2×D412yes—
173C7×C4⋊C4—22no—
174C7×C4⋊C4C22×C412yes—
175C7×C22⋊C4—21no—
176C7×C22⋊C4—22no—
177C42×C7—22no—
178C42×C7—21no—
179C42×C7C22×C412yes—
180C42×C7C2×D411yescharacteristic
181C42×C7C2×D411yescharacteristic
182C22⋊Dic7—21no—
183#1144—22no—
184#1143—22no—
185#1142—22no—
186#1141—22no—
187(C2×D7)⋊C4—22no—
188C4⋊Dic7C22×C412yes—
189Dic7⋊C4—22no—
190C4×Dic7—22no—
191C4×Dic7—21no—
192C4×Dic7—21no—
193C4×Dic7—21no—
194C4×Dic7#4112yes—
195C4×Dic7#4112yes—
196#1136—21no—
197#1136—21no—
198#1136—21no—
199#1136—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8C22×C412yes—
205C2×C7⋊C8C22×C411yescharacteristic
206C2×C7⋊C8C22×C411yescharacteristic
207C2×C7⋊C8#4112yes—
208C2×C7⋊C8#4112yes—
209C2×#135—71no—
210C4×#41—71no—
211C4×#41C2×D711yescharacteristic
212#442—72no—
213#441—72no—
214#440—72no—
215#440—72no—
216#433—22no—
217#433—142no—
218#431—22no—
219#431—142no—
220C2×#108—71no—
221C2×#108—71no—
222#427—71no—
223#424—71no—
224#424—71no—
225#424C2×D711yescharacteristic
226#424C2×D711yescharacteristic
227C2×#104—71no—
228C2×#104—71no—
229C2×#104—71no—
230C2×#104—71no—
231#416—71no—
232#416—71no—
233#408—71no—
234#408—71no—
235#408C2×D711yescharacteristic
236#408C2×D711yescharacteristic
237C4×M16—71no—
238C4×M16C2×D711yescharacteristic
239C2×#97—71no—
240C2×#97—71no—
241C2×#97—71no—
242C2×C4×C8—71no—
243C2×C4×C8—71no—
244C2×C4×C8C2×D711yescharacteristic
245#376—71no—
246#341—71no—
247#341—71no—
248#339—71no—
249#339—71no—
250#339—71no—
251#339—71no—
252C23×C7—21no—
253C23×C7C2×#4111yescharacteristic
254C22×D7—21no—
255C7×Q8—21no—
256C7×Q8—21no—
257C7×D4—21no—
258C7×D4—21no—
259C7×D4—41no—
260C7×D4—41no—
261C2×C4×C7—22no—
262C2×C4×C7—21no—
263C2×C4×C7—42no—
264C2×C4×C7—44no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7C4×D412yes—
268C2×C4×C7C4×D412yes—
269C2×C4×C7C23×C411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C22⋊D7—41no—
276C22⋊D7—41no—
277C2×Dic7—21no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283D28—21no—
284D28—21no—
285C4×D7—44no—
286C4×D7—41no—
287C4×D7—41no—
288Dic14—21no—
289Dic14—21no—
290C7×C8—22no—
291C7×C8—22no—
292C7×C8—41no—
293C7×C8—41no—
294C7⋊C8—22no—
295C7⋊C8—41no—
296C7⋊C8—41no—
297C7⋊C8#11712yes—
298C7⋊C8#11712yes—
299C2×#41—71no—
300C2×#41C22×D711yescharacteristic
301#135—142no—
302#135—142no—
303C2×Q16—71no—
304C2×SD16—71no—
305C2×SD16—71no—
306C2×D8—71no—
307C2×M16—71no—
308C2×M16—71no—
309C2×M16C22×D711yescharacteristic
310C2×M16C22×D711yescharacteristic
311C22×C8—71no—
312C22×C8—71no—
313C22×C8C22×D711yescharacteristic
314C22×C8C22×D711yescharacteristic
315C4×Q8—72no—
316C4×Q8C4×D712yes—
317C4×D4—22no—
318C4×D4—72no—
319C4×D4—142no—
320C4×D4C4×D712yes—
321#117—21no—
322#117—71no—
323#117—141no—
324#117C22×D711yescharacteristic
325C2×C42—21no—
326C2×C42—71no—
327C2×C42—71no—
328C2×C42—141no—
329C2×C42C22×D711yescharacteristic
330#108—142no—
331#108—142no—
332#107—72no—
333#106—72no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—142no—
337C4⋊C8—142no—
338#104—142no—
339#104—142no—
340#104—142no—
341#104—142no—
342#103—72no—
343#103—72no—
344#102—72no—
345#102—72no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—142no—
349C22⋊C8—142no—
350#97—21no—
351#97—72no—
352#97—72no—
353#97—142no—
354#97—142no—
355#97—141no—
356C4×C8—21no—
357C4×C8—142no—
358C4×C8—142no—
359C4×C8—141no—
360C4×C8C22⋊D712yes—
361C4×C8C22⋊D712yes—
362#95—142no—
363C22×C7—22no—
364C22×C7—42no—
365C22×C7—41no—
366C22×C7C2×C4×D411yescharacteristic
367C22×C7#52911yescharacteristic
368C22×C7#52911yescharacteristic
369C2×D7—42no—
370C2×D7—41no—
371C4×C7—22no—
372C4×C7—22no—
373C4×C7—42no—
374C4×C7—41no—
375C4×C7C2×#11711yescharacteristic
376C4×C7C4×#4111yescharacteristic
377C4×C7#53211yescharacteristic
378C4×C7#53211yescharacteristic
379Dic7—42no—
380Dic7—42no—
381Dic7—42no—
382Dic7—41no—
383#41—21no—
384#41—21no—
385#41—21no—
386#41—21no—
387#41—141no—
388#41—141no—
389#41—141no—
390#41—141no—
391C2×Q8—71no—
392C2×Q8C2×C4×D711yescharacteristic
393C2×D4—21no—
394C2×D4—71no—
395C2×D4—141no—
396C2×D4C2×C4×D711yescharacteristic
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—142no—
402C22×C4—141no—
403C22×C4#115712yes—
404C22×C4C23×D711yescharacteristic
405Q16—142no—
406SD16—142no—
407SD16—142no—
408D8—142no—
409M16—21no—
410M16—21no—
411M16—21no—
412M16—21no—
413M16—141no—
414M16—141no—
415M16—141no—
416M16—141no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—71no—
424C2×C8—71no—
425C2×C8—72no—
426C2×C8—72no—
427C2×C8—72no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8C2×C22⋊D712yes—
433C2×C8C2×C22⋊D712yes—
434C4⋊C4—22no—
435C4⋊C4—72no—
436C4⋊C4—142no—
437C4⋊C4C2×C4×D712yes—
438C22⋊C4—21no—
439C22⋊C4—22no—
440C22⋊C4—141no—
441C22⋊C4—142no—
442C42—22no—
443C42—21no—
444C42—72no—
445C42—72no—
446C42—142no—
447C42—141no—
448C42—141no—
449C42—141no—
450C42C2×C4×D712yes—
451C42C2×C22⋊D711yescharacteristic
452C42C2×C22⋊D711yescharacteristic
453C2×C7—21no—
454C2×C7—21no—
455C2×C7—81no—
456C2×C7#229311yescharacteristic
457C2×C7#295512yes—
458D7—81no—
459C23—21no—
460C23—141no—
461C23C2×#115711yescharacteristic
462Q8—21no—
463Q8—21no—
464Q8—141no—
465Q8—141no—
466D4—21no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—141no—
471D4—141no—
472D4—281no—
473D4—281no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—142no—
481C2×C4—142no—
482C2×C4—142no—
483C2×C4—141no—
484C2×C4—284no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—281no—
488C2×C4—281no—
489C2×C4C4×C22⋊D712yes—
490C2×C4C4×C22⋊D712yes—
491C2×C4C22×C4×D711yescharacteristic
492C2×C4D7×#4111yescharacteristic
493C2×C4D7×#4111yescharacteristic
494C2×C4D7×#4111yescharacteristic
495C2×C4D7×#4111yescharacteristic
496C2×C4C22×C22⋊D711yescharacteristic
497C8—22no—
498C8—22no—
499C8—41no—
500C8—41no—
501C8—72no—
502C8—72no—
503C8—142no—
504C8—281no—
505C8—281no—
506C7#2129611yescharacteristic, complemented, Sylow
507C22—22no—
508C22—41no—
509C22—42no—
510C22—281no—
511C22—282no—
512C22#9021911yescharacteristic
513C22#9021911yescharacteristic
514C22C2×C4×C22⋊D711yescharacteristic
515C4—22no—
516C4—22no—
517C4—41no—
518C4—42no—
519C4—282no—
520C4—282no—
521C4—281no—
522C4—282no—
523C4D7×#11711yescharacteristic
524C4#9021311yescharacteristic
525C4#9052311yescharacteristic
526C4#9052311yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—561no—
531C2#1173967011yescharacteristic
532C2#1174123712yes—
subgroupG/H
centerC2×C4C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2922C2
socleC22×C7C2×C4×D4
prime radicalC2×#6513C22
cosocleC22×C4×C7C24
hypercenter#2922D7
hypocenterC7#21296
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2922

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C7×#2922C2semidirect
3. #11735900C2semidirect
4. #11729397C2semidirect
5. #11728292C2semidirect
6. #11728291C2semidirect
7. #11728290C2semidirect
8. #11728289C2semidirect
11. #11727752C2semidirect
12. #11727751C2semidirect
13. #11727627C2semidirect
14. #11727620C2semidirect
15. #11727503C2semidirect
42. #89802C4semidirect
43. #89798C4semidirect
44. #89790C4semidirect
45. #89786C4semidirect
126. #2922D7semidirect
506. C7#21296semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#29222
#117399512
#117359002
#117293972
#117282922
#117282912
#117282902
#117282892
#117277542
#117277532
#117277522
#117277512
#117276272
#117276202
#117275032
#212967